Math 101 Exam 3

Transformations (continued)

• Vertical/Horizontal Stretches and compressions: If $k>1$,
  • $f(x)=kg(x)$ vertically stretches $g(x)$ by a factor of $k$
  • $f(x)=\frac{1}{k}g(x)$ vertically compresses the graph of $g(x)$ by a factor of $k$
  • $f(x)=g(kx)$ horizontally compresses $g(x)$ by a factor of $k$
  • $f(x)=g(\frac{1}{k}x)$ horizontally stretches the graph of $g(x)$ by a factor of $k$
• Combining Transformations: Order matters when we apply multiple types of vertical transformations or multiple types of odd transformations:
  • To determine the order of vertical transformations, use PEMDAS
  • To determine the order of horizontal transformations, use SADMEP
  • You can deal with vertical transformations then horizontal, or vice-versa, the order between the two types doesn't matter.

Quadratic Functions

• A quadratic function $f(x)$ is a function that can be written in standard form: $f(x)=ax^2+bx+c$, where $a,b,$ and $c$ are constants and $a\neq 0$.
  • Observe that $f(0)=a(0)^2+b(0)+c=c$, so $c$ is the output value of the $y$-intercept. Hence, standard form is useful for finding the $y$-intercept.
• The factored form of a quadratic (if it exists) is $f(x)=a(x-r)(x-s)$, where $a\neq 0$ and $r$ and $s$ are the $x$-intercepts.
  • Sometimes, the $x$-intercepts can be hard to find. This is why it's handy to have the quadratic formula: $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$ This can be handy to indicate there are no $x$-intercepts if $\sqrt{b^2-4ac}$ is not possible to find.
  • If a function is in standard form, you can put it into factored form by using the $ac$-method
• The vertex form of a quadratic function is $f(x)=a(x-h)^2+k$, where $a\neq 0$, $(h,k)$ is the coordinate of the vertex of the parabola, and $x=h$ is the axis of symmetry for the parabola.
  • You can view this as a transformation of the funcion $g(x)=x^2$, where we've shifted horizontally by $h$, vertically by $k$, and stretched or compressed our graph vertically by some factor.
  • To get vertex form from standard form, use completing the square.

Combining Functions

• Adding, subtracting, dividing, multiplying, scaling, and composing are all ways we can combine functions together
• When determining the units of a combination of two functions, study the units of each function

Power Functions

• A power function can be written in the form $f(x)=kx^p$, where $k$ and $p$ are constants.
  • Just because it doesn't look like a power function, doesn't mean it can't be written as one. For example, $f(x)=\frac{6}{\sqrt{4t}}$ is a power function since $f(x)=\frac{6}{(4t)^{1/2}}=\frac{6}{2t^{1/2}}=3t^{-1/2}$. Here, $k=3$ and $p=-1/2$.
• Graphs of $x^p$: See texed version. Use chapter 5 content to sketch graphs of $kx^p$.

Polynomial Functions

• A polynomial function is a function that can be written as a sum of power functions whose exponents are non-negative integers. So a polynomial is a combination of special types of power functions.
  • So $f(x)=x^4+\pi x^2+3$, linear, and quadratic functions are all examples of polynomials. Functions with negative exponents or exponents that are not whole numbers are not polynomials.
  • When a polynomial is written as a sum of power functions, it is in standard form. (Think about the standard form for quadratics.)
  • Each power function $kx^p$ is called a term.
  • The leading term is the term with the largest exponent.
  • The degree is the exponent of the leading term
• The long-run behavior of a polynomial is the same as the long-run behavior of its leading term. But it's leading term is a power function... so we can use our graphs of $x^p$. (!!)
• We can talk about the short-run behavior of a polynomial if it's in a factored form. Something like $f(x)=2(x-3)^2(x-1)(x+4)^5$, where we have a product of linear polynomials raised to some power.
  • A function bounces off of the $x$-axis at zeroes with even multiplicities
  • A function crosses through the $x$-axis at zeroes with odd multiplicities
  • The sum of the multiplicities ends up being the degree of the polynomial. So the above polynomial has degree 8, and so the leading term is $2x^8$.

And although not on exam 3, the following doesn't really merit it's own page, so it's included here...

Rational Functions

• A rational function $r(x)$ can be written as $r(x)=\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials, and $q(x)\neq 0$. So a rational function is a combination of polynomials.
  • Some simplification may be necessary to show a function is a rational function
• The long run behavior of a rational function $r(x)=\frac{p(x)}{q(x)}$ is the same as the long run behavior of the leading term of $p(x)$ divided by the leading term of $q(x)$. This will be a power functions, so we can (again!) use our graphs of $x^p$. (!!)
  • For example, if $r(x)=\frac{x^3+x^2+1}{3x^4+3}$, the long run behavior of $r(x)$ is the same as the long run behavior of $\frac{x^3}{3x^4}=\frac{1}{3}x^{-1}$.
• The interesting short run behavior we're interested in involves holes, vertical asymptotes, and zeroes. If $r(x)=\frac{p(x)}{q(x)}$ is a rational function in simplified form,
  • The zeroes of $r(x)$ are the same as the zeroes of $p(x)$.
  • The vertical asymptotes of $r(x)$ are the same as the zeroes of $q(x)$.
  • The holes come from the version of $r(x)$ that is not in simplified form -- it's the zeroes that $q(x)$ and $p(x)$ both have in common. These correspond to the factors you cancelled to get $r(x)$ into simplified form.